One could summarize the theme of random geometry as the analysis of random metric structures on manifolds (typically surfaces). It’s a research topic situated at the interface of probability, geometry and combinatorics, with strong connections to statistical systems and quantum field theory in physics. In this talk I will highlight the random planar map approach to random geometry, involving the combinatorial analysis of discrete metric spaces arising from large random graphs in the plane. A key result in this area is the appearance of the Brownian sphere, a strongly curved random metric on the 2-sphere, as the universal scaling limit of random planar map models like the uniform random planar quadrangulation. Rather different limiting behavior appears when one imposes curvature restrictions, particularly when requiring the random metrics to be flat. Some recent progress on this front will be discussed.